Advertisements
Advertisements
Question
If tan \[\alpha = \frac{1}{1 + 2^{- x}}\] and \[\tan \beta = \frac{1}{1 + 2^{x + 1}}\] then write the value of α + β lying in the interval \[\left( 0, \frac{\pi}{2} \right)\]
Advertisements
Solution
\[\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}\]
\[ = \frac{\frac{1}{1 + 2^{- x}} + \frac{1}{1 + 2^{x + 1}}}{1 - \frac{1}{(1 + 2^{- x} )(1 + 2^{x + 1} )}}\]
\[ = \frac{1 + 2^{x + 1} + 1 + 2^{- x}}{1 + 2^{x + 1} + 2^{- x} + 2^{- x + x + 1} - 1}\]
\[ = \frac{2 + 2^{x + 1} + 2^{- x}}{2 + 2^{x + 1} + 2^{- x}}\]
\[ = 1\]
\[\text{ Therefore }, \alpha + \beta = \tan^{- 1} (1) = \frac{\pi}{4} .\]
APPEARS IN
RELATED QUESTIONS
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Find the value of: sin 75°
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/2`
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
sin (A − B)
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
sin (A + B)
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
cos (A + B)
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that:
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
Prove that:
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
If \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , then show that \[\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\].
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
tan 3A − tan 2A − tan A =
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
If sinθ + cosθ = 1, then find the general value of θ.
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2
[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
The value of tan3A - tan2A - tanA is equal to ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.
If sinx + cosx = a, then |sinx – cosx| = ______.
